2.2 Dynamics Of Rigid Bodies.pdf 381f63

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Dynamics and control of mechanical systems Date

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Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation, angular velocity vector, description of velocity and acceleration in relatively moving frames. Euler angles, Review of methods of momentum and angular momentum of system of particles, inertia tensor of rigid body. Dynamics of rigid bodies - Euler equations, application to motion of symmetric tops and gyroscopes and problems of system of bodies. Kinetic energy of a rigid body, virtual displacement and classification of constraints. D’ Alembert’s principle. Introduction to generalized coordinates, derivation of Lagrange's equation from D’ Alembert’s principle. Small oscillations, matrix formulation, Eigen value problem and numerical solutions. Modelling mechanical systems, Introduction to MATLAB®, computer generation and solution of equations of motion. Introduction to complex analytic functions, Laplace and Fourier transform. PID controllers, Phase lag and Phase lead compensation. Analysis of Control systems in state space, pole placement, computer simulation through MATLAB.

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Content Purpose: Focus on 4 Preview of dynamics of rigid bodies 4 Euler equations,

to motion of symmetric tops gyroscopes and problems of system of bodies.

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4 Application

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and

Introduction The dynamics of rigid bodies can be analyzed as 4 Rotation about a fixed point*

åM

O

dH O dt dH O = + W x H O If rotating about a secondary dt coordinate (not body fixed) =

where H = å r x m (w x r ) OR 4 General 3D motion (about center of mass of the body) O

i

i

i

dH dt dH = + WxH dt where H = å Ri x mi (w x Ri )

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åM =

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Dynamics of rigid body 4 For a moving coordinate system x-y-z with angular

velocity Ω, the moment relation becomes:

å M = (H% )

xyz

+ WxH

% i+H % j +H % k) + W xH = (H x y z #"! #$ $$" $$$ ! Change in magnitude of H

Change in direction of H

Show that this is correct!

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where

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Dynamics of rigid body 4 For reference axes attached to the body, moments and

products of inertia will become invariant with time, 4 And Ω = ω 4 In general (rotation about a fixed axis)

åM

Ang. velocity of rotating frame attached to body

! ) fixed = (H ! )body + w x H ; = (H C C C

C

axis

æ d ( Iw ) ö =ç ÷ è dt ø = Iw! = (I1w!1 )i + (I 2w! 2 )j

axis

body C axis

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C

= external applied torque

w x H = - (w2w3 (I 2 - I 3 ))i - (w3w1 (I 3 - I1 ))j - (w1w2 (I1 - I 2 ))k

(H! )

+ (I 3w! 3 )k

åM

I1w!1 - w2w3 (I 2 - I 3 ) = M x ü ï I 2w! 2 - w3w1 (I 3 - I1 ) = M y ý Euler ' s Equations ï I 3w! 3 - w1w2 (I1 - I 2 ) = M z þ

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Euler’s Equations 4 For a reference axes coinciding with the principal axes

of inertia The origin can be either at the mass center (C) or at a point O fixed to the body and fixed in space The products of inertia: Ixy = Iyz = Ixz = 0

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Thus, the moment equation becomes

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Dynamics of rigid body 4 When no external torque is applied on a rigid body,

angular momentum is conserved dH T T = 0 where H = Iw Þ [H1 H 2 H 3 ] = [I1w1 I 2w2 I 3w3 ] dt Where the subscripts are simplified as follows: xx = 1, yy = 2 and zz = 3

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The Euler equations for this case (principal axis, no external torque)*

I1w!1 = w2w3 (I 2 - I 3 ) I 2w! 2 = w3w1 (I 3 - I1 )

Show that these are correct!

I 3w! 3 = w1w2 (I1 - I 2 ) DMS6021 - Dynamics and Control of Mechanical Systems

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Dynamics of rigid body… Euler Equations 4 Euler’s equations are the 3D equations of motion for a rigid

body - used to analyze the motion of a rigid body 4 Using the three components of Newton’s 2nd law and Euler

equations, motion of a rigid body in 3D is completely defined 4 Three steps of rigid body analysis using Euler equations: 1. Choose a coordinate system (that rotates about a fixed point O or

that has its origin at the center of mass) 2. Draw the free body diagram 3. Apply the equations of motion (Newton’s second law and equations JU – 2017/HL

of angular motion)

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Dynamics of rigid body… Euler Equations 4 Please read and try to understand about (Not pensum)

Application of the relations of rigid body motion to

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motion of symmetric tops and gyroscopes

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Summary and Questions 4 Considered dynamics of rigid body Rotation about a fixed axis Rotation about the center of mass Moment equation for a moving coordinate system

4 Derived Euler´s Equations for

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motion with moving coordinate system a coordinate system attached to the rotating body cases when no external torque is applied and cases when the axis of rotation coincides with axis principal axes of inertia

?

Next: Kinetic energy of a rigid body, virtual displacement and classification of constraints DMS6021 - Dynamics and Control of Mechanical Systems

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Exercise 1 4 A small particle of mass m and its restraining cord are spinning with an

angular velocity ω on the horizontal surface of a smooth disk, shown in section view below. As the force Fs is slowly increased, r decreases and ω changes. Initially, the mass is spinning with ω0 and r0. Determine: i) an expression for ω as a function of r, and ii) the work done on the particle by Fs between r0 and an arbitrary r. the principle of work and energy.

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Solution

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Exercise 2 4 A pendulum consisting of a mass, M, is suspended by a rigid rod of

length L. The pendulum is initially at rest and the mass of the rod can be neglected. A bullet of mass m and velocity v0 impacts M and stays embedded in it. The angle that the velocity vector v0 forms with the horizontal is α. Find out the angle θmax reached by the pendulum.

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Solution

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